Spinners Jack Leach and Dom Bess bowled Somerset to the brink of a crucial victory.
Left-armer Leach followed up his five first-innings wickets with four for 79 from 47 overs as the visitors, who began the day on 28 without loss following on, progressed to 247 for eight by stumps - a lead of just 45.
Bess provided good support on a pitch offering only slow turn and had figures of three for 85 from 38 overs at the close. Craig Overton was the other wicket-taker.
There were half-centuries from Haseeb Hameed and Liam Livingstone in a more convincing Lancashire batting effort. But from 144 for one, their score was another disappointment.
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The visitors extended their opening partnership to 82 before a short rain break disrupted Alex Davies' concentration. The third ball after the resumption saw him offer a return catch to Leach, aiming through the leg side, having made 49.
It was 89 for one when another shower forced an early lunch. On a pitch offering little assistance to seam or spin, Somerset's bowlers had their work cut out.
Hameed reached a studious 151-ball half-century, with six fours, but on 62 drove Leach to cover where Tom Abell took a straightforward catch.
Steven Croft fell cheaply to Bess, lbw trying to sweep a full toss, and the young off-spinner went on to remove Shiv Chanderpaul for nine, bowling the veteran West Indies star through the gate.
At tea Lancashire were 175 for four and required a further 27 to make Somerset bat again. The final session saw the hosts take the second new ball and Overton broke through with it as Dane Vilas, on 14, feathered a catch behind.
Livingstone went to an excellent fifty, made off 153 balls, with three fours and a six, before falling on 57.
He was given out caught behind off Leach, although wicketkeeper Davies removed the bails and appeared to appeal to the square-leg umpire for a stumping.
Ryan McLaren battled hard for his 15 runs but was then snapped up off bat and pad by Marcus Trescothick at short-leg to give Leach his fourth wicket.
Bess then had Stephen Parry caught at silly mid-off stretching forward for 12 before Ryan Jarvis and Tom Bailey ensured the game went into a fourth day.
Somerset could move out of the First Division relegation zone with victory on Friday.